Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $X/\mathbb{C}$ a nonsingular proper variety and $X_{an}$ it's associated analytic space. Is $X_{an}$ necessarily Kahler? Certainly we know this if $X$ is projective.

A complex torus is algebraic iff it is projective. Are there Kahler manifolds which are algebraic, but not projective?

share|improve this question
    
A smooth proper complex variety is Kaehler iff it is projective. –  ulrich Sep 28 '12 at 4:12
    
@ulrich: Thanks! –  LMN Sep 28 '12 at 12:28

1 Answer 1

up vote 9 down vote accepted

Any abstract algebraic compact complex manifold is Moishezon. By Moishezon's theorem, any Kähler Moishezon manifold is projective algebraic. There are non-projective proper complex varieties, so $X_{an}$ is not necessarily Kähler. This is represented in the diagram at the end of Hartshorne's Algebraic Geometry Appendix B.

In summary, all of your questions have negative answers.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.